Stable group theory and approximate subgroups

Details Stable group theory and approximate subgroups Stable group theory and approximate subgroups

2009-09-11

arxiv.org/abs/0909.2190v4

Mathematics

Logic

Further local corrections, thanks to two anonymous referres

Scientific paper

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |X X \^{-1} X |/ |X| bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Combining these methods with Gromov's proof, we show that a finitely generated group with an approximate subgroup containing any given finite set must be nilpotent-by-finite. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.

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